where \(\circ \)  is a composition operator, \(v\)  is the velocity field (\(v = z\) ) and \(\phi ^{(0)}=Id\)  is an identity transformation. We follow  [1]}, [2]}, [3]}, [4]} to integrate the SVF \(v\)  over time \(t=[0,1]\)  by a scaling and squaring layer (SS) to obtain the final motion field \(\phi ^{(1)}\)  at time \(t=1\) . Specifically, starting from \( \phi ^{(1/2^{T})}=p+v(p)/2^{T}\)  where \(p\)  is a spatial location, by using the recurrence \( \phi ^{(1/2^{t})}= \phi ^{(1/2^{t+1})}\circ \phi ^{(1/2^{t+1})}\)  we can compute \( \phi ^{(1)}= \phi ^{(1/2)}\circ \phi ^{(1/2)}\) . In our experiments, \(T=7\) , which is chosen so that \(v(p)/2^{T}\)  is small enough.
With the latent variable \(z\) , we can compute the motion field \(\phi \)  by the SS layer. We then use a spatial transform layer to warp image \(x\)  by \(\phi \)  and we obtain a noisy observation of the warped image, \(x\circ \phi \) , which could be a Gaussian distribution:
\(p(y|z;x)=\mathcal {N}(y;x\circ \phi ,\sigma ^{2}\mathbb {I}),\) 
